# What's a Transitive Group Action?

/Let a group $G$ act on a set $X$. The action is said to be transitive if for any two $x,y\in X$ there is a $g\in G$ such that $g\cdot x=y$. This is equivalent to saying there is an $x\in X$ such that $\text{orb}(x)=X$, i.e.

*there is exactly one orbit*. And all this is just the fancy way of saying that $G$ shuffles all the elements of $X$*among themselves*. In other words,**
"What happens in $X$ stays in $X$."**

*none*of the marbles in the little box fell*outside*the box and onto the floor ($Y$). Even though the elements of $X$ were shuffled, they remained within $X$.